Topic C: Dividing Larger Numbers by One-Digit Divisors
Several methods are used to divide larger numbers. This book will only teach one method. If you have learned a different method for dividing, ask your instructor to review it with you. You can use the practice exercises in this workbook using whichever method you prefer.
One-Digit Divisors without Remainders
Division has four steps which are repeated until the dividend is completely divided. Work through the three examples which show these steps.
- Divide.
- Multiply
- Subtract and compare the remainder to the divisor.
- Bring down the next digit in the dividend and repeat.
Example A
| 294 ÷ 7 = | |
| Rewrite as [latex]7\enclose{longdiv}{294}[/latex] | |
| Step 1: Divide.
Find a trial quotient using multiplication tables or division facts. Look at the dividend one digit at a time.
The first number in the quotient is 4. Place the 4 in the quotient directly above the 9 tens. The 4 is 4 tens in the quotient. |
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\end{array}[/latex] |
| Step 2: Multiply.
Multiply 4 × 7 = 28. Write the 28 under the 29. Draw a line. |
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\28\hspace{0.8em}\\\hline\end{array}[/latex] |
| Step 3: Subtract.
Subtract 29 − 28 = 1 (ten). Check 1 < 7. |
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline1\hspace{0.8em}\end{array}[/latex] |
| Step 4: Bring down the next number in the dividend.
Bring down the next number in the dividend (4), and put it beside the result of the subtraction you just completed (1). Together, they make 14. This 14 is the number that you must now divide. |
[latex]\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\end{array}[/latex] |
Repeat Step 1: Divide.
|
[latex]\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\end{array}[/latex] |
Repeat Step 2: Multiply.
|
[latex]\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\\14\\\hline\end{array}[/latex]
|
Repeat Step 3: Subtract.
|
[latex]\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\\-14\\\hline0\end{array}[/latex] |
There are no more numbers in the dividend to bring down, so you are done.
294 ÷ 7 = 42
Example B
128 ÷ 2 =
[latex]\begin{array}{r}64\\2 \enclose{longdiv}{128}\\-12\hspace{0.8em}\\\hline08\\-8\\\hline0\end{array}[/latex]
Step 1: Divide.
- Can 2 go into 1? No.
- Can 2 go into 12? Yes.
- How many times?
- 2 × 6 = 12; 12 ÷ 2 = 6
- The first number in the trial quotient is 6. Put the 6 in the quotient directly above the 2 tens dividend.
Step 2: Multiply.
- 6 × 2 = 12.
- Write the 12 under the 12. Draw a line.
Step 3: Subtract.
- 12 − 12 = 0
- Check 0 < 12
Step 4: Bring down the next number in the dividend.
- Bring down the next number in the dividend (8), and put it beside the result of the subtraction you just completed (0).
- 8 is now the number to be divided.
Repeat
- Divide: 8 ÷ 2 = 4
- Multiple: 4 × 2 = 8
- Subtract: 8 − 8 = 0
- Check 0 < 2
- Bring down the next digit. There are no more digits in the dividend.
128 ÷ 2 = 64
Exercise 1
Find the quotients (divide, multiply, subtract, compare). Check your work using the answer key at the end of the exercise.
- [latex]4\enclose{longdiv}{364}[/latex]
- [latex]2\enclose{longdiv}{144}[/latex]
- [latex]5\enclose{longdiv}{455}[/latex]
- [latex]7\enclose{longdiv}{651}[/latex]
- [latex]8\enclose{longdiv}{144}[/latex]
- [latex]2\enclose{longdiv}{166}[/latex]
- [latex]7\enclose{longdiv}{588}[/latex]
- [latex]2\enclose{longdiv}{196}[/latex]
- [latex]5\enclose{longdiv}{230}[/latex]
- [latex]8\enclose{longdiv}{584}[/latex]
- [latex]6\enclose{longdiv}{366}[/latex]
- [latex]4\enclose{longdiv}{244}[/latex]
- [latex]5\enclose{longdiv}{375}[/latex]
- [latex]8\enclose{longdiv}{200}[/latex]
- [latex]2\enclose{longdiv}{628}[/latex]
- [latex]7\enclose{longdiv}{357}[/latex]
- [latex]9\enclose{longdiv}{837}[/latex]
- [latex]8\enclose{longdiv}{248}[/latex]
- [latex]8\enclose{longdiv}{312}[/latex]
- [latex]7\enclose{longdiv}{462}[/latex]
- [latex]5\enclose{longdiv}{295}[/latex]
- [latex]6\enclose{longdiv}{384}[/latex]
- [latex]2\enclose{longdiv}{276}[/latex]
- [latex]4\enclose{longdiv}{372}[/latex]
Answers to Exercise 1
- 91
- 72
- 91
- 93
- 18
- 83
- 84
- 98
- 46
- 73
- 61
- 61
- 75
- 25
- 314
- 51
- 93
- 31
- 39
- 66
- 59
- 64
- 138
- 93
Example C
856 ÷ 8 =
[latex]8\enclose{longdiv}{856}[/latex]
| Divide
Does 8 go into 8? Yes. 8 ÷ 8 = 1. |
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\end{array}[/latex] |
| Multiply
1 × 8 = 8 |
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\8\hspace{1.2em}\end{array}[/latex] |
|
|
|
| Subtract
8 − 8 = 0. Check 0 < 8. |
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline0\hspace{1.2em}\end{array}[/latex]
|
| Bring down the next digit
5 is now the number to be divided. |
[latex]\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\end{array}[/latex] |
Now repeat.
| Divide
Does 8 go into 5? No. You must put a zero to hold the place in the quotient. If a digit is brought down, a digit must be placed in the quotient. |
[latex]\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\end{array}[/latex] |
| Multiply
0 × 8 = 0 |
[latex]\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline\hspace{0.8em}\end{array}[/latex] |
|
|
|
| Subtract
5 − 0 = 5. Check 5 < 8. |
[latex]\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline5\hspace{0.8em}\end{array}[/latex]
|
| Bring down the next digit
56 is now the number to be divided. |
[latex]\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\end{array}[/latex] |
Now repeat.
| Divide
Does 8 go into 56? Yes. 56 ÷ 8 = 7. Write 7 in the quotient in the ones place above the 6 in the dividend. |
[latex]\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\end{array}[/latex] |
| Multiply
7 × 8 = 56 |
[latex]\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\\56\end{array}[/latex] |
| Subtract
56 − 56 = 0. Check 0 < 8.
|
[latex]\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\\-56\\\hline0\end{array}[/latex] |
| Bring down the next digit.
No more digits. |
856 ÷ 8 = 107
Exercise 2
Find the quotients (divide, multiply, subtract, compare). Check your work using the answer key at the end of the exercise.
- [latex]7\enclose{longdiv}{721}[/latex]
- [latex]9\enclose{longdiv}{954}[/latex]
- [latex]3\enclose{longdiv}{927}[/latex]
- [latex]3\enclose{longdiv}{621}[/latex]
- [latex]4\enclose{longdiv}{824}[/latex]
- [latex]9\enclose{longdiv}{972}[/latex]
- [latex]7\enclose{longdiv}{714}[/latex]
- [latex]2\enclose{longdiv}{416}[/latex]
- [latex]5\enclose{longdiv}{540}[/latex]
- [latex]6\enclose{longdiv}{654}[/latex]
- [latex]8\enclose{longdiv}{832}[/latex]
- [latex]4\enclose{longdiv}{436}[/latex]
- [latex]5\enclose{longdiv}{515}[/latex]
- [latex]2\enclose{longdiv}{814}[/latex]
- [latex]6\enclose{longdiv}{648}[/latex]
- [latex]8\enclose{longdiv}{856}[/latex]
- [latex]5\enclose{longdiv}{525}[/latex]
- [latex]7\enclose{longdiv}{763}[/latex]
- [latex]9\enclose{longdiv}{945}[/latex]
- [latex]3\enclose{longdiv}{315}[/latex]
- [latex]8\enclose{longdiv}{872}[/latex]
- [latex]4\enclose{longdiv}{416}[/latex]
- [latex]6\enclose{longdiv}{618}[/latex]
- [latex]2\enclose{longdiv}{612}[/latex]
Answers to Exercise 2
- 103
- 106
- 309
- 207
- 206
- 108
- 102
- 208
- 108
- 109
- 104
- 109
- 103
- 407
- 108
- 107
- 105
- 109
- 105
- 105
- 109
- 104
- 103
- 306
One-Digit Divisors with Remainders
Do the division exactly the same way that you have been learning. Often there is a remainder after the last subtraction. Write it with the quotient as you already know how to do.
Example D
259 ÷ 8 =
[latex]\begin{array}{l}\hspace{1.5em}32\text{ R}3\\8 \enclose{longdiv}{259}\\\hspace{0.3em}-24\\\hline\hspace{1.7em}19\\\hspace{1em}-16\\\hline\hspace{2.3em}3\end{array}[/latex]
259 ÷ 8 = 32 R3
Exercise 3
Divide and show any remainders. Check your work using the answer key at the end of the exercise.
- [latex]2\enclose{longdiv}{93}[/latex]
- [latex]3\enclose{longdiv}{52}[/latex]
- [latex]5\enclose{longdiv}{94}[/latex]
- [latex]7\enclose{longdiv}{74}[/latex]
- [latex]4\enclose{longdiv}{95}[/latex]
- [latex]9\enclose{longdiv}{96}[/latex]
- [latex]6\enclose{longdiv}{97}[/latex]
- [latex]8\enclose{longdiv}{99}[/latex]
- [latex]9\enclose{longdiv}{98}[/latex]
- [latex]4\enclose{longdiv}{59}[/latex]
- [latex]6\enclose{longdiv}{76}[/latex]
- [latex]3\enclose{longdiv}{79}[/latex]
- [latex]7\enclose{longdiv}{96}[/latex]
- [latex]5\enclose{longdiv}{57}[/latex]
- [latex]2\enclose{longdiv}{47}[/latex]
- [latex]8\enclose{longdiv}{91}[/latex]
- [latex]7\enclose{longdiv}{89}[/latex]
- [latex]6\enclose{longdiv}{82}[/latex]
- [latex]5\enclose{longdiv}{67}[/latex]
- [latex]2\enclose{longdiv}{85}[/latex]
- [latex]4\enclose{longdiv}{71}[/latex]
- [latex]3\enclose{longdiv}{65}[/latex]
- [latex]9\enclose{longdiv}{92}[/latex]
- [latex]8\enclose{longdiv}{94}[/latex]
Answers to Exercise 3
- 46 R1
- 17 R1
- 18 R4
- 10 R4
- 23 R3
- 10 R6
- 16 R1
- 12 R3
- 10 R8
- 14 R3
- 12 R4
- 26 R1
- 13 R5
- 11 R2
- 23 R1
- 11 R3
- 12 R5
- 13 R4
- 13 R2
- 42 R1
- 17 R3
- 21 R2
- 10 R2
- 11 R6
Check Division by Multiplying
To check your division, do this:
- Multiply the quotient by the divisor.
- Add on any remainder.
The product will equal the dividend if your arithmetic is correct.
Example E
[latex]\begin{array}{l}\hspace{1.5em}52\text{ R}1\\9 \enclose{longdiv}{469}\\\hspace{0.3em}-45\\\hline\hspace{1.7em}19\\\hspace{1em}-18\\\hline\hspace{2.3em}1\end{array}[/latex] [latex]\begin{array}{rr}&52\\\times&9\\\hline&468\end{array}[/latex] [latex]468+1=469[/latex]
Exercise 4
Divide and check your answer by multiplying. Check your work using the answer key at the end of the exercise.
- [latex]7\enclose{longdiv}{709}[/latex]
- [latex]2\enclose{longdiv}{423}[/latex]
- [latex]5\enclose{longdiv}{538}[/latex]
- [latex]4\enclose{longdiv}{609}[/latex]
- [latex]9\enclose{longdiv}{406}[/latex]
- [latex]6\enclose{longdiv}{125}[/latex]
- [latex]3\enclose{longdiv}{605}[/latex]
- [latex]9\enclose{longdiv}{928}[/latex]
- [latex]3\enclose{longdiv}{962}[/latex]
- [latex]4\enclose{longdiv}{805}[/latex]
- [latex]8\enclose{longdiv}{301}[/latex]
- [latex]2\enclose{longdiv}{807}[/latex]
- [latex]6\enclose{longdiv}{725}[/latex]
- [latex]7\enclose{longdiv}{320}[/latex]
- [latex]9\enclose{longdiv}{140}[/latex]
- [latex]8\enclose{longdiv}{483}[/latex]
- [latex]2\enclose{longdiv}{197}[/latex]
- [latex]6\enclose{longdiv}{307}[/latex]
- [latex]5\enclose{longdiv}{504}[/latex]
- [latex]8\enclose{longdiv}{709}[/latex]
- [latex]7\enclose{longdiv}{876}[/latex]
- [latex]4\enclose{longdiv}{101}[/latex]
- [latex]3\enclose{longdiv}{269}[/latex]
- [latex]5\enclose{longdiv}{473}[/latex]
Answers to Exercise 4
- 101 R2
- 211 R1
- 107 R3
- 152 R1
- 45 R1
- 20 R5
- 201 R2
- 103 R1
- 320 R2
- 201 R1
- 37 R5
- 403 R1
- 120 R5
- 45 R5
- 15 R5
- 60 R3
- 98 R1
- 51 R1
- 100 R4
- 88 R5
- 125 R1
- 25 R1
- 89 R2
- 94 R3
Topic C: Self-Test
Mark /24 Aim 19/24
A. Find the quotient. (6 marks)
- [latex]6\enclose{longdiv}{96}[/latex]
- [latex]4\enclose{longdiv}{92}[/latex]
- [latex]7\enclose{longdiv}{91}[/latex]
- [latex]2\enclose{longdiv}{93}[/latex]
- [latex]5\enclose{longdiv}{94}[/latex]
- [latex]3\enclose{longdiv}{52}[/latex]
B. Divide. (6 marks)
- [latex]7\enclose{longdiv}{182}[/latex]
- [latex]8\enclose{longdiv}{736}[/latex]
- [latex]6\enclose{longdiv}{162}[/latex]
- [latex]5\enclose{longdiv}{295}[/latex]
- [latex]4\enclose{longdiv}{184}[/latex]
- [latex]9\enclose{longdiv}{576}[/latex]
C. Divide and show your check for each answer (12 marks – 1 mark for question, 1 mark for check).
- [latex]9\enclose{longdiv}{705}[/latex]
- [latex]4\enclose{longdiv}{257}[/latex]
- [latex]7\enclose{longdiv}{899}[/latex]
- [latex]5\enclose{longdiv}{538}[/latex]
- [latex]8\enclose{longdiv}{876}[/latex]
- [latex]6\enclose{longdiv}{628}[/latex]
Answers to Topic C Self-Test
- Find the quotient.
- 16
- 23
- 13
- 46 R1
- 18 R4
- 17 R1
- Divide.
- 26
- 92
- 27
- 59
- 46
- 64
- Divide and show your check for each answer.
- 78 R3
- 64 R1
- 128 R3
- 107 R3
- 109 R4
- 104 R4